In the metric distortion problem there is a set of candidates and a set of voters, all residing in the same metric space. The objective is to choose a candidate with minimum social cost, defined as the total distance of the chosen candidate from all voters. The challenge is that the algorithm receives only ordinal input from each voter, in the form of a ranked list of candidates in non-decreasing order of their distances from her, whereas the objective function is cardinal. The distortion of an algorithm is its worst-case approximation factor with respect to the optimal social cost. A series of papers culminated in a 3-distortion algorithm, which is tight with respect to all deterministic algorithms. Aiming to overcome the limitations of worst-case analysis, we revisit the metric distortion problem through the learning-augmented framework, where the algorithm is provided with some prediction regarding the optimal candidate. The quality of this prediction is unknown, and the goal is to evaluate the performance of the algorithm under a accurate prediction (known as consistency), while simultaneously providing worst-case guarantees even for arbitrarily inaccurate predictions (known as robustness). For our main result, we characterize the robustness-consistency Pareto frontier for the metric distortion problem. We first identify an inevitable trade-off between robustness and consistency. We then devise a family of learning-augmented algorithms that achieves any desired robustness-consistency pair on this Pareto frontier. Furthermore, we provide a more refined analysis of the distortion bounds as a function of the prediction error (with consistency and robustness being two extremes). Finally, we also prove distortion bounds that integrate the notion of $\alpha$-decisiveness, which quantifies the extent to which a voter prefers her favorite candidate relative to the rest.

翻译：在度量失真问题中，存在一组候选人和一组选民，所有个体均处于同一度量空间。目标是选择一位使社会成本最小的候选人，社会成本定义为所选候选人与所有选民之间的距离总和。挑战在于算法仅能从每位选民处获得序数输入，即按与选民距离非递减顺序排列的候选人排名列表，而目标函数却是基数形式的。算法的失真度是其相对于最优社会成本的最坏情况近似比。一系列研究最终提出了一个3-失真算法，这一结果对于所有确定性算法而言是紧的。为了克服最坏情况分析的局限性，我们通过学习增强框架重新审视度量失真问题，在该框架中，算法会获得关于最优候选人的某种预测。该预测的质量未知，目标是在预测准确时评估算法的性能（称为一致性），同时对于任意不准确的预测仍提供最坏情况保证（称为鲁棒性）。作为主要结果，我们刻画了度量失真问题中鲁棒性与一致性之间的帕累托前沿。我们首先识别出鲁棒性与一致性之间不可避免的权衡关系，进而设计了一族学习增强算法，能够实现该帕累托前沿上的任意期望的鲁棒性-一致性组合。此外，我们提供了作为预测误差函数的失真界更精细分析（其中一致性和鲁棒性为两个极端情况）。最后，我们还证明了整合$\alpha$-决定性概念的失真界，该概念量化了选民对其首选候选人的偏好程度相对于其他候选人的差异。